# MSSM flat direction inflation: slow roll, stability, fine tunning and reheating

###### Abstract:

We consider low scale slow roll inflation driven by the gauge invariant flat directions udd and LLe of the Minimally Supersymmetric Standard Model at the vicinity of a saddle point of the scalar potential. We study the stability of saddle point and the slow roll regime by considering radiative and supergravity corrections. The latter are found to be harmless, but the former require a modest finetuning of the saddle point condition. We show that while the inflaton decays almost instantly, full thermalization occurs late, typically at a temperature GeV, so that there is no gravitino problem. We also compute the renormalization group running of the inflaton mass and relate it to slepton masses that may be within the reach of LHC and could be precisely determined in a future Linear Collider experiment.

## 1 Introduction

Recently we have constructed a model of inflation [1] based
on the udd and LLe flat directions of Minimally
Supersymmetric Standard Model (MSSM; for a review of MSSM flat
directions, see [2]). In this model the inflaton is a
gauge invariant combination of either squark or slepton fields. For
a choice of the soft SUSY breaking parameters and the inflaton
mass , the potential along the flat udd and LLe
directions is such that there is a period of slow roll inflation of
sufficient duration to provide the observed spectrum of CMB
perturbations. In the inflationary part of the MSSM potential the
second derivative is vanishing and the slow roll phase is driven by
the third derivative of the potential ^{1}^{1}1In a recent similar
model with small Dirac neutrino masses, the observed microwave
background anisotropy and the tilted power spectrum are related to
the neutrino masses [3]. The model relies solely on
renormalizable couplings..

MSSM inflation occurs at a very low scale with GeV and with field values much below the Planck scale. Hence it stands in strong contrast to the conventional inflation models which are based on ad hoc gauge singlet fields and often employ field values close to Planck scale (for a review, see [4]). In such models the inflaton couplings to SM physics are unknown. As a consequence, much of the post-inflationary evolution, such as reheating, thermalization, generation of baryon asymmetry and cold dark matter, which all depend very crucially on how the inflaton couples to the (MS)SM sector [5, 6, 7, 8], is not calculable from first principles. The great virtue of MSSM inflation based on flat directions is that the inflaton couplings to Standard Model particles are known and, at least in principle, measurable in laboratory experiments such as LHC or a future Linear Collider.

However, as in almost all inflationary models, a fine tuning of the initial condition is needed to place the flat direction field to the immediate vicinity of the saddle point at the onset of inflation. In addition, there is the question of the stability of the saddle point solution and of the existence of a slow roll regime. These are issues that we wish to address in detail in the present paper. Both supergravity and radiative corrections to the flat direction inflaton potential must be considered. Hence we need to write down and solve the renormalization group (RG) equations for the MSSM flat directions of interest. RG equations are also needed to scale the model parameters, such as the inflaton mass, down to TeV scale; since the inflaton mass is related either to squark or slepton masses, it could be measured by LHC or a future Linear Collider.

Because the inflaton couplings to ordinary matter are known, inflaton decay and thermalization are processes that can be computed in an unambiguous way. Unlike in many models with a singlet inflaton, in MSSM inflation the potential relevant for decay and thermalization cannot be adjusted independently of the slow roll part of the potential.

This paper is organized as follows. In Sect. 2 we present the model of MSSM inflation and its predictions. In Sect. 3 we study the flat direction potential without an exact saddle point. We find generic constraints for the existence of a slow roll solution and show that in the slow roll regime there is always tunneling from a false minimum. In Sect. 4 we solve the one-loop RG equations for the LLe flat direction and find the one-loop corrected saddle point. We quantify the amount of fine tuning required for the slow roll solution to exist, and relate through RG running the LLe inflaton mass with observables such as the slepton masses at the LHC energy scale. We also show that supergravity corrections to the potential can be neglected. In Sect. 5 we discuss the decay of the flat direction, the reheating and thermalization of the Universe, and show that the reheat temperature is low enough for the model to avoid the gravitino problem. Sect. 6 contains our conclusions and some discussion about future prospects.

## 2 The Model

Let us recapitulate the main features of MSSM flat direction
inflation [1]. As is well known, in the limit of unbroken
SUSY the flat directions have exactly vanishing potential. This
situation changes if we take into account soft SUSY breaking and
non-renormalizable superpotential terms ^{2}^{2}2Our framework is
MSSM together with gravity, so consistency dictates that all
non-renormalizable terms allowed by gauge symmetry and supersymmetry
should be included below the cut-off scale, which we take to be the
Planck scale. Some interseting issues on A-term inflation were also
discussed in Ref. [10]. of the type [2]

(1) |

where is a
superfield which contains the
flat direction. Within MSSM all the flat directions are lifted by
non-renormalizable operators with [11], where
depends on the flat direction. We expect that quantum gravity effects yield
GeV and [12] ^{3}^{3}3Note however that our results will be valid
for any values of , because rescaling simply shifts
the VEV of the flat direction..

Let us focus on the lowest order superpotential term in Eq. (1) which lifts the flat direction. Soft SUSY breaking induces a mass term for and an -term so that the scalar potential along the flat direction reads

(2) |

Here and denote respectively the radial and the
angular coordinates of the complex scalar field
, while is the phase of the
-term (thus is a positive quantity with dimension of mass).
Note that the first and third terms in Eq. (2) are positive
definite, while the -term leads to a negative contribution along
the directions whenever . ^{4}^{4}4The
importance of the A-term has also been highlighted in a successful
MSSM curvaton model [13].

In principle, in the -term all the superpotential terms of a given dimension may enter with a different coefficient ; whether they are related or not depends on the details of the SUSY breaking mechanism.

### 2.1 The Saddle Point

The maximum impact from the -term is obtained when (which occurs for values of ). Along these directions has a secondary minimum at (the global minimum is at ), provided that

(3) |

At this minimum the curvature of the potential is positive both along
the radial and angular directions^{5}^{5}5If the is too large,
the secondary minimum will be deeper than the one in the origin, and
hence becomes the true minimum. However, this is phenomenologically unacceptable as such a minimum will break charge
and/or color [12]. with .

As discussed in [1], if the local minimum is too steep, the field will become trapped there with an ensuing inflation that has no graceful exit like in the old inflation scenario [14]. On the other hand in an opposite limit, with a point of inflection, a single flat direction cannot support inflation [15].

However, in the gravity mediated SUSY breaking case, the -term and the soft SUSY breaking mass terms are expected to be the same order of magnitude as the gravitino mass, i.e.

(4) |

Therefore, as pointed out in [1], in the gravity mediated SUSY breaking it is possible that the potential barrier actually disappears and the inequality in Eq. (3) is saturated so that and are related by

(5) |

This represents a fine tuning and will be discussed at length in the next Sections. However, let us now assume for the sake of argument that Eq. (5) holds. Then both the first and second derivatives of vanish at , i.e. . As the result, if initially , a slow roll phase of inflation is driven by the third derivative of the potential.

Note that this behavior does not seem possible for other SUSY
breaking scenarios such as the gauge mediated breaking [16] or
split SUSY [17]. In split SUSY the -term is protected by an
-symmetry, which also keeps the gauginos light while the sfermions
are quite heavy [17] ^{6}^{6}6In the gauge mediated case
there is an inherent mismatch between and , except at
very large field values where Eq. (4) can be satisfied.
However there exists an unique possibility of a saddle point inflation
which we will discuss separately [18]..

### 2.2 Slow roll

The potential near the saddle point Eq. (5) is very flat along the real direction but not along the imaginary direction. Along the imaginary direction the curvature is determined by . Around the field lies in a plateau with a potential energy

(6) |

with

(7) |

This results in Hubble expansion rate during inflation which is given by

(8) |

When is very close to , the first derivative is extremely small. The field is effectively in a de Sitter background, and we are in self-reproduction (or eternal inflation) regime where the two point correlation function for the flat direction fluctuation grows with time. But eventually classical friction wins and slow roll begins at [1]

(9) |

The slow roll potential in this case reads

(10) |

We can now solve the equation of motion for the field in the slow-roll approximation,

(11) |

assuming initial conditions such that the flat direction starts in the vicinity of with . Inflation ends when the slow roll parameter, becomes of . This occurs at

(12) |

which happens to be also the place when the other slow roll paremeter becomes of .

The number of e-foldings during the slow roll from to is given by

(13) |

where we have used (this is justified since ), and Eq. (11). The total number of e-foldings in the slow roll regime is then found from Eq.(9)

(14) |

The observationally relevant perturbations are generated when . The number of e-foldings between and , denoted by follows from Eq. (13)

(15) |

The amplitude of perturbations thus produced is given by

(16) |

where we have used Eqs.(8), (2.2), (15). Again after using these equations, the spectral tilt of the power spectrum and its running are found to be

(17) | |||

(18) |

### 2.3 Properties and predictions

As discussed in [1], among the about 300 flat directions there are two that can lead to a successful inflation along the lines discussed above.

One is udd which, up to an overall phase factor, is parameterized by

(19) |

Here are color indices, and denote the quark families. The flatness constraints require that and .

The other direction is LLe, parameterized by (again up to an overall phase factor)

(20) |

where are the weak isospin indices and denote the lepton families. The flatness constraints require that and . Both these flat directions are lifted by non-renormalizable operators,

(21) |

The reason for choosing either of these two flat
directions^{7}^{7}7Since LLe are udd are independently
- and -flat, inflation could take place along any of them but
also, at least in principle, simultaneously. The dynamics of multiple
flat directions are however quite involved [19]. is twofold:
(i) a non-trivial -term arises, at the lowest order, only at ;
and (ii) we wish to obtain the correct COBE normalization of the CMB
spectrum.

Those MSSM flat directions which are lifted by operators with dimension are such that the superpotential term contains at least two monomials, i.e. is of the type

(22) |

If represents the flat direction, then its VEV induces a large effective mass term for , through Yukawa couplings, so that . Hence Eq. (22) does not contribute to the -term.

More importantly, as we will see, all other flat directions except those lifted by fail to yield the right amplitude for the density perturbations. Indeed, as can be seen in Eq. (7), the value of , and hence also the energy density, depend on .

According to the arguments presented above, successful MSSM flat direction inflation has the following model parameters:

(23) |

Here we assume that (we drop the subscript ”6”) is of order one, which is the most natural assumption when .

The Hubble expansion rate during inflation and the VEV of the saddle
point are ^{8}^{8}8We note that and depend
very mildly on as they are both .

(24) |

Note that both the scales are sub-Planckian. The total energy density stored in the inflaton potential is . The fact that is sub-Planckian guarantees that the inflationary potential is free from the uncertainties about physics at super-Planckian VEVs. The total number of e-foldings during the slow roll evolution is large enough to dilute any dangerous relic away, see Eq. (14):

(25) |

Domains which are initially closer than to , see Eq. (9), can enter self-reproduction in eternal inflation, with no observable consequences.

At such low scales as in MSSM inflation the number of e-foldings, , required for the observationally relevant perturbations, is much less than [20]. If the inflaton decays immediately after the end of inflation, we obtain . Despite the low scale, the flat direction can generate adequate density perturbations as required to explain the COBE normalization. This is due to the extreme flatness of the potential (recall that ), which causes the velocity of the rolling flat direction to be extremely small. From Eq. (16) we find an amplitude of

(26) |

There is a constraint on the mass of the flat direction from the amplitude of the CMB anisotropy:

(27) |

We get a lower limit on the mass parameter when . For smaller values of , the mass of the flat direction must be larger. Note that the above bound on the inflaton mass arises at high scales, i.e. . However, through renormalization group flow, it is connected to the low scale mass, as will be discussed in Sect. 4.

The spectral tilt of the power spectrum is not negligible because,
although the first slow roll parameter is , the other slow roll parameter is given by
and thus, see
Eq. (17)^{9}^{9}9Obtaining (or ,
which is however outside the allowed region) requires
deviation from the saddle point condition in Eq. (5), see
Section 3. For a more detailed discussion on the spectral tilt, see
also Refs. [9],[21].

(28) | |||

(29) |

where we have taken (which is the maximum value allowed for the scale of inflation in our model). In the absence of tensor modes, this agrees with the current WMAP 3-years’ data within [22]. Note that MSSM inflation does not produce any large stochastic gravitational wave background during inflation. Gravity waves depend on the Hubble expansion rate, and in our case the energy density stored in MSSM inflation is very small.

## 3 Sensitivity of the saddle point inflation

In previous Sections and in Ref. [1] the dynamics of the flat direction inflaton was discussed assuming the saddle point condition Eq. (5) is satisfied exactly. The question then is, how large a deviation can be allowed for before slow roll inflation will be spoiled. There are obviously two distinct possibilities: either or . (Although we always take in the present paper, we keep here for generality of the formalism.) In the former case there is a barrier which separates the global minimum and the false minimum at . The eventual inflationary trajectory starts near the top of the barrier. The field can either start at the top, or jump to its vicinity from the false minimum via Coleman-de Luccia tunneling [23]. As we will see, if the barrier is too high, there will be no inflation near the top. In the latter case there is no minimum but the potential may be too steep for slow roll inflation. Therefore we need to analyze the two cases separately. However, the steepness of the potential is a problem which is common to both cases and is addressed at the end of this Section.

To facilitate the discussion, let us define

(30) |

Here we will assume that . Before beginning the calculations, we would like to point that the main results of this section are summarized in Fig. (3) and Eq. (60). These yield no constraint on the spectral tilt as any value consistent with sufficient slow roll inflation (i.e. a number of e-foldings) is allowed.

### 3.1 The potential for

In this case there are two extrema, a maximum and a minimum ,

(31) |

and a point of inflection

(32) |

We can then express the potential and its derivatives at the extrema and as functions of ,

(33) | |||

(34) | |||

(35) |

Note that when , so we can expand the potential around the maximum, and include the small correction due to deviations from the saddle point as

(36) |

The maximum is now at , and the minimum is at , with masses (curvature of the potential) given by , which coincides with Eq. (34), while there is now a point of inflection at . Note that the difference in potential height between the maximum and the minimum is

(37) |

In the limit of we recover the saddle point. We will work in the limit when .

Let us now define a few variables, , and . Then the equation of motion for the scalar field down the potential can be written as

(38) |

where we have used (36).

The eventual inflationary trajectory will start in the vicinity of the maximum and will roll down the hill towards . The field can either start near the maximum, or tunnel to its vicinity out of the false vacuum. Tunneling takes place in the presence of a non-zero vacuum energy, , and is known as Coleman-de Luccia tunneling [23].

In order to find the interpolating solution between the false and the true vacuum one solves the Euclidean equation of motion,

(39) |

whose exact solution is

(40) |

This solution starts at and ends at . The “tunneling” from to can actually be understood as diffusion due to de Sitter fluctuations. It is valid so long as . This requires that, see Eq. (34),

(41) |

This also insures that at the maximum, and hence inflation can take place after tunneling. Otherwise there will be no inflation, neither in self-reproduction nor in slow roll regime.

Let us now discuss the effect of the tunneling solutions on the tilt of the CMB spectrum. Again there is self-reproduction close to the maximum as long as the curvature there is smaller than the rate of expansion squared, i.e. . The slow-roll regime starts at when

(42) |

Note that , see Eq. (33), and . We therefore find

(43) |

Now we integrate Eq. (38) in the slow-roll approximation, using a new variable , for which the equation of motion becomes . The exact solution is, in terms of the number of -folds, ,

(44) |

where we have defined

(45) |

Note that in the limit , we recover the usual expression Eq. (13). From Eqs. (12), (43), (44), it turns out that the number of e-folds from to the end of inflation at is again of order .

The required number of e-folds for the relevant perturbations () determines the value of ,

(46) |

On the other hand, the amplitude of fluctuations is given by

(47) |

which for becomes

(48) | |||

while the spectral tilt and its running are universal,

(49) | |||

(50) |

which reduce to the usual expressions in the limit , see Eqs. (17), (18). We show in Fig. 3 the variation of the tilt with for a model with . Note that the range of allowed values of is constrained by the condition to have inflation near the maximum, i.e. that at . This gives , for , see Eqs. (41,45). The corresponding range of tilt values agrees with the results of Ref. [9].

### 3.2 The potential for

When , instead of a saddle point we have a point of inflection at , where . We find

(51) |

and

(52) | |||

(53) | |||

(54) |

The slow-roll parameters at the point of inflection are

(55) | |||

(56) |

Unlike the previous cases, (saddle point), and (tunneling solution), there is no point (except the origin ) where . This implies that there will be no self-reproduction regime unless [9].

However this is not troublesome as long as we have a sufficient number of e-foldings, arising due to a slow roll inflation.

The amplitude and tilt of the scalar spectrum in the case can be obtained from the analytical continuation of the results of previous subsection (),

(57) | |||

(58) | |||

(59) |

which is in agreement with the results of Ref. [9].

The dependence of the tilt on can be seen in Fig. 3. Note that, as pointed out in Ref. [9], the tilt can get any value in the allowed range of which is determined by the viability of slow roll. In the future we will have to determine what value of agrees with observations.

To summarize the fine tuning issue, for typical values of GeV ^{10}^{10}10The tendency from radiative corrections is to
raise , see Section 4.3., the saddle point condition,
Eq. (5), requires fine-tuning at the level of

(60) |

which is not negligible.

## 4 Radiative and supergravity corrections

The MSSM inflaton candidates are represented by gauge invariant combinations but are not singlets. The inflaton parameters receive corrections from gauge interactions which, unlike in models with a gauge singlet inflaton, can be computed in a straightforward way. Quantum corrections result in a logarithmic running of the soft supersymmetry breaking parameters and . One might then worry about their impact on Eq. (5) and the success of inflation.

In this section we will discuss running of the potential with VEV-dependent values of and in Eq. (5). Our conclusion is that the running of the gauge couplings do not spoil the existence of a saddle point. However the VEV of the saddle point is now displaced; by how much will depend precisely on the inflaton candidate. In order to discuss the situation, we derive a general expression for the one-loop effective potential for the flat directions, and then focus on the direction, for which the system of RG equations can be solved analytically.

### 4.1 One-loop effective potential

The first thing to check is whether the radiative corrections remove the saddle point altogether. The object of interest is the effective potential at the phase minimum , for which we obtain

(61) | |||||

where , , and are the values of , and given at a scale . Here is chosen to be real and positive (this can always be done by re-parameterizing the phase of the complex scalar field ), and are coefficients determined by the one-loop renormalization group equations.

Our aim is to find a saddle point of this effective potential, so we calculate the 1st and 2nd derivatives of the potential and set them to zero. This is a straightforward although somewhat cumbersome exercise that results in the expression

(62) |

where , , and are values of the parameters at the scale . Inserting this into , we can then find the condition to have a saddle point at :

(63) |

In the limit when , this mercifully simplifies to

(64) | |||

(65) |

Note that Eqs. (4.1), (64) give the necessary relations between the values of and as calculated at the saddle point. The coefficients need to be solved from the renormalization group equations at the scale given by the saddle point . Since are already one loop corrections, taking the tree-level saddle point value as the renormalization scale is sufficient.

Hence we may conclude that, although the soft terms and the value of the saddle point are all affected by radiative corrections, they do not remove the saddle point nor shift it to unreasonable values. The existence of a saddle point is thus insensitive to radiative corrections.

### 4.2 RG equations for the direction

The form of the relevant RG equations depend on the flat direction. RG equations for are simpler since only the gauge interactions are involved and the lepton Yukawa couplings are negligible. The case of requires numerics if is chosen from the third family. For other choices, however, it closely resembles . For the one-loop RG equations governing the running of , , and with the scale are given by [24]

(66) |

Here , denote the mass of the
and gauginos respectively and are the associated
gauge couplings. It is a straightforward exercise to obtain the
equations that govern the running of and associated with
the superpotential term (which lifts the flat direction). Note that has the same
quantum numbers as , and hence in this respect
combination behaves just like . One can
then use the familiar RG equations that govern the Yukawa coupling and
-term associated with the superpotential
term [24]. However, as explained in [25], the
coefficients of the terms on the right-hand side are proportional to
the number of superfields contained in a superpotential
term. ^{11}^{11}11We would like to thank Manuel Drees for explaining this
point to us. Hence the second and third equations in (4.2) are
simply obtained from those for the term after multiplying
by a factor of . The first equation in (4.2) is also easily
found by taking the electroweak charges of , and
superfields into account while taking into account that .

The running of gauge couplings and gaugino masses obey the usual equations [24]:

(67) |

The solutions of the renormalization group equations are

(68) | |||||

(69) | |||||

(70) | |||||

(71) | |||||

(72) |

where , and . Ignoring the running of the gaugino masses and gauge couplings, we find that

(73) |

where the subscript denotes the values of parameters at the high scale .

For universal boundary conditions, as in minimal grand unified supergravity, the high scale is the GUT scale GeV, and , . Then we just use RG equations to run the coupling constants and masses to the scale of the saddle point GeV for GeV, TeV, . With these values we obtain

(74) | |||||

(75) | |||||

(76) |

where is calculated at the GUT scale.

Typically the running based on gaugino loops alone results in negative values of [26]. Positive values can be obtained when one includes the Yukawa couplings, practically the top Yukawa, but the order of magnitude remains the same.